Slide rule for belleville springs

ABSTRACT

This invention provides a mechanically simple slide rule which will solve the following equations.

United States Patent [72] Inventor Joseph W. Wells 56] References Cit d Brim" UNlTED STATES PATENTS 746594 3 257 075 6/1966 Na m 11 235/70 22, y a [45] Patented Mar. 16, 1971 Primary Examiner-Stephen J. Tomsky [73] Assignee Associat d Spring Corpor t n Att0rney-Scrivener, Parker, Scrivener and Clarke Bristol, Conn.

ABSTRACT: This invention provides a mechanically simple slide rule which will solve the following equations. 54 sum; RULEFOR BELLEVILLE SPRINGS P= [(h-f) (h-f/Z) 3] (1) 4 Claims, 11 Drawing Figs. "Ef a 52 U.S.Cl 235/70 &,= a (h-f/2)+C t] 2 [51 Int. Cl G06g 1/02 (1W2) 2 50 Field of Search 235/70, Ef 1 70.1 7m

,2 LOAD i & FLAT 1.0mm? I? "?l,.m 5 ?7PPl,9 4 5 F 747N i. a; a fimavxm on 09 6/ e's'w s ""2 L665; .5"A"' .2

L'OOO BID PATENTEU NARI 6 I97! ATTOR NEYS PMENTEU 7m 7 6197! SHEET 3 UF 5 OZI OLI

OBI

OEI

OLI

O O O n O O 0 Q 0 DEFLECTION IN PERCENTAGE OFh INVENTOR.

JOSEPH W. WELLS ATTORNEYS SHEET [1F 5 ATTORNEYS PATENTED MARI s |97| DESCRIPTION OF THE DRAWINGS FIG. 1A is a diametrical sectional view of a Belleville spring washer;

FIG. 1B is a view of part of the washer shown in FIG. 1A;

FIG. 2 is an elevational view of part of a conventional standard slide rule;

FIG. 3 is an elevational view of the slide rule of FIG. 2.with an inverted scale;

FIG. 4 is an elevational view of part of a conventional slide rule in which each cycle of scale A is one-half as long as scale D, and each cycle of scale K is one-third as long as scale D;

FIG. 5 is an elevational view of a conventional slide rule having one slide and five scales, one of which is inverted;

FIG. 6 shows load-deflection curves for use with the computer provided by the invention, for determining intermediate loads when the load at flat is known;

FIG. 7 is an elevational view of the thickness side of the computer;

FIG. 8 is the same as FIG. 7 with the slide inverted;

FIG. 9 is an elevational view of the stress side of the com puter; and

FIG. 10 is a graph showing variation in the load at flat with the OD/ID ratio.

DEFINITION OF TERMS The following nomenclature is used throughout this specification, reference being made in part to the illustration of a Belleville washer in FIGS. 1A and 1B:

R=0D/ID a=0D/2, in inches c=percent of deflection to flat .E.=Modulus of elasticity 30,000,000 for steel) =MateriaI factor (10 is omitted on the slide rule) The derivations of formulas and the logic of ,the invention will be clarified by an explanation of asimple slide rule. FIG. 2

illustrates a conventional rule having two fixedside parts and .one slide. Any two adjacent scales on the rule and the slide provide for the solution of the problem xy=z. This is accomplished by the graphic addition of the logarithms or:

FIG. 2 illustrates a standard slide rule which is adjusted so that x=2, y=4 and z=8 and solves the equation 2 4=8. This also illustrates division by the graphic subtraction of logarithms. In FIG. 2, 84x

If one of the scales is inverted so that values increase from right to left the same operationsmay be performed by dealing with the reciprocal of one of the factors, e.g., 2X4=;;

the equation 2Xbeing illustrated by scale CI in FIG. 3.

Powers and roots of numbers are handled by varying the proportions of the scales. In FIG. 4 each cycle of scale A is half as long as scale D and each cycle of scale K is one-third as long as scale D, and the setting illustrates the equations 2 x4 and 23X 8. In a conventional slide rule scales are aligned with each other, but in a special slide rule one of the scales is offset from the others, this being equivalent to introducing a constant into DESCRIPTION OF THE INVENTION All of the foregoing is common practice in the design of standard and special slide rules. Others have coped with complicated formulas by adding extra slides, rotatingarms and variable scales, but these devices are expensive to build and require special training in their use. In contrast, my invention consists in theingenious manipulation of the formulas so that complicated formulas may be solved on a simple slide rule with the aid of one chart. One of the principal objects of the invention is to focus the designers attention on the ratio of height to thickness which determinesthe shape of the loaddeflection curves as shown in FIG. 6. Others have introduced arbitrary factors to aid in the solution of the formulas (see for example, Society of Automotive Engineers Handbook Supplement, HS 63). Graphic representations of these factors conceal the basic shape of the load-deflection curves because they include elements of OD/ID and f/t. In my invention h/t is a principal factor in the transpositions of all three basic equations. In describing the computer we will derive the mathematical equations first and then explain the placement of the scales on the computer.

The first step in the simplificationof equation (1) is due to the observation that the formula for load at flat position (when f==h) eliminates (h-f), (h-f/Z) at t, resulting in the following equation:

Ehi

The next step is the observation that loads at intermediate deflections depend upon the ratio of height to thickness as shown by the following: Divide equation (I) by equation (4). This gives percent of intermediate load to load at flat.

P f f f) (ii- H1 Now let c=percent of deflection to flaFf/t. Then f=ch and P load factor 15/ 1 -0 material factor h height factor I thickness factor M ratio of OD/ID factor a outside diameter factor I reduce this to four variables and introduce wanted factors as follows:

This is an important element of invention because it introduces the ratio of height to thickness which relates to FIG.

30 By substituting steel values for the material factors and (OD/ID=2a becomes a constant and equation (6) has the four variables and a constant required to fit a simple slide rule.

Refer now to FIG. 7 which is the thickness $19 of the computer. Scale 1 is P in equation (6), scale 2 is 0D in equation (6), scale 3 is q in equation (6) and scale 4 is t in equation (6). Scales 1, 2, and 4 are aligned according to their indices. Scale 3 is located by solving formula 6 for any assumed values.

For example: P =100, OD=1, q=1

The length of one cycle of each scale is determined by the exponent attached to that factor in equation (6). P, and q having the samggxponent, one, scales 1 and 3 have the same base length. OD having the exponent 2, each cycle of scale 2 is twice as long and I having the exponent 4, each cycle of scale 4 is four times the base scale length. This principle is illustrated in FIG. 4. The inverse direction of scales 2 and 3 will be discussed later.

Referring to equation (4,it is apparent that P for any other material than steel would be proportional to the revised value of the material factor scales on the slide rule are common logarithms. The value of M assumed in order to locate scale 3 was 0.685 for a ratio OD/ID=2. Other typical values, M, for other ratios of OD/ID and the corresponding correction factors are as follows:

OD/ID M M/M 1.2 0.291 1 2.35 1.3 0.388 1 1.76 1.5 0.523 1.31 4. 0.775 0.89

The adjusting scale for this factor is marked 6 in FIG. 7. Since the factor M has the same exponent as the P scale, the intervals are the same. Scale 6 is inverted for a reason which will be discussed later, but the logic of its construction may be seen in FIG. 8 in which the slide has been inverted; this shows the agreement of the scale with the above table. Scale 7 is identical to scale 6. It is repeated so that one scale or the other will be visible under the cursor regardless of the position of the slide.

STRESS SIDE Equation (2) is a stress-deflection formula. To make it compatible with the load-thickness side, it is necessary to combine equations (land (2in order to relate load at flat to stress at flat. This provides the common denominator of load at flat between the two sides of the rule.

It is common practice in spring design technique to combine load deflection and load stress formulas. This is usually done by dividing or multiplying one formula by the other. This is not applicable to equations (land (2) because dividing would eliminate too many of the variables and multiplying would produce an exceedingly complex equation. In accordance with the invention I combine the formulas by substitution, retaining the ratio of height to thickness and other required variables in a relatively simple form.

By substitution ofqt for h equation (4) may be written:

Let le i, (c1q+2c2) w/E simplify Scr '\/PF cF then B (10) This is the form of the basic slide rule as illustrated in FIG. 5, except for the fact that q, the factor for h/: cannot be isolated from C and C which depends upon OD/ID. The scale for q, therefore, cannot be a simple logarithmic scale. This would normally be considered an insurmountable obstacle to the construction of a simple slide rule.

I solve this problem by substituting basic values in equation 10) as follows.

Referring now to FIG. 9, scales 8, 9 and 10 are located and spaced in accordance with principles previously discussed. Scale 11 is spaced according to equation (l1) and located with S =275,000 aligned with h/t=l as shown in the above example.

Scale I2 is the same as scale 5 on the thickness side.

Scale 13 was established in the same way as scale 6 on the thickness side, but it is much more complex because equation (9) contains C and C as well as M. The complete equation and a graph of the function are shown in FIG. 10. This scale is labeled below the line for increasing values of P as OD/ID varies from 1.2 to 1.7 and above the line for decreasing values as OD/ID varies from 1.7 to 4.0. 1.7 is the peak of the curve in FIG. 10.

Scale 14 is the same as scale 13.

Equation (3) is the formula for stress at the point S, in FIG. 1B. Scales 15, I6 and 17 correspond for this equation to scales 11, 13 and 14 for equation (2) and they were derived, spaced and located in a similar manner.

DIRECTION OF SCALES For ease of use it is desirable to locate similar scales in the same sequence on both sides of the rule. This I do as follows.

Thickness Stress Side Side Standard Scale (Figure 5):

Load fiat Load fiat. CIIIIIIIII 'Il 11/t.II.'III h/t Thickness." Stress.

On the stress side of the rule, equation I) is in the form LLB B C and, therefore, the scales read from right to left the same as FIG. 5. On the thickness side the equation (6) is in the form AB=CD. I put this in the desired form by writing equation (6) This requires that scales 2 and 3 be inverted and multiplication is accomplished by division by the reciprocal as illustrated in FIG. 3.

Each of the auxiliary scales, 5. 6, 7, I2, l3, 14, 16 and 17 has an arrow to indicate the standard value on which the rule is based without adjustment. For example, scale has an arrow at 33 which is the material factor for steel. An adjustment may be made either by (a) setting the hairline over the arrow and moving the slide to the new value or (b) setting the hairline over the new value and moving the slide to the arrow. The direction of the scales has been established so that procedure (a) will be used when progressing from the lower to the upper main scales and procedure (b) will be used when progressing from the upper main scales to the lower.

SPECIAL NOTATIONS On scale 6 there is a dot between the digits 2 and 4 which represents 2.5. This notation is used because the function is changing very rapidly at this point. Standard notation would call for a line at 3, but this would be practically superimposed on 4.

On the stress side of the slide rule there is a red circle which overlaps scales 8 and 9 on the extreme right. This is to remind the operator to replace the slide so that it faces correctly.

I claim:

1. A slide rule for use in designing a Belleville spring, comprising two spaced elongated fixed members, a slide between said members and coextensive therewith, and a cursor mounted on the fixed members for movement along them. the parts of the rule having two sides on which there are the following scales:

a. On one side of the rule scales for determining thickness in inches according to the formula:

q PF= ZFD in which P =load at flat position E=modulus of elasticity (30,000,000 for steel) o-=Poissons ratio h=inside height, in inches t=thickness OD=outside diameter, in inches,

in which formula steel values are substituted for material values and 0D/ID=2 so that 01 becomes a constant and oz=l .93 I08 one of these scales representing values of P from I to 10,000 extending from end to end of one of the fixed members, a second scale representing values of 0D from 10 down to 0.1 on the slide adjacent and coterminous with the P scale, a third scale representing values of I from 0.001 to 1.0 inch on the second fixed member and coterminous with P and OD scales, a fourth scale on the slide representing values of q from 1.0 to 3 in the direction of 0.010 on the t scale and from 1.0 to 2 in the opposite direction and with the 1.0 value of q aligned with 0.0268 on the t scale if it is assumed that P,.-=l00, 0D=l and q=l and at least one scale on the slide representing values of OD/ID from 1.2 to 4,

b. and on the other side of the rule scales for determining the stress at flat position according to the following formula:

5 S cF OD T a in which P =load at flat position OD=outside diameter ScF=compressive stress on convex side of the inner edge of flat, in p.s.i.

E=modulus of elasticity (30,000,000) for steel r=Poissons ratio second fixed member and coterminous with the P scale, a

fourth scale on the slide adjacent the ScF scale representing values of ill! from 1.0 down to 0.2 in the direction of the lower end of the ScF scale and from 1.0 to 3.0 in the other direction, with 1.0 on the h/t scale at 275,000 on the ScF scale when the 50F and P scales are coterminous and when it is assumed that the terms of the B equation have the following values:

E=l0 l0 0'=0.3

2. A slide rule according to claim 1, comprising in addition at least one scale on the stress side of the slide representing values of OD/ID from 1.2 to 4, for use in determining stress OD/ID is other than 2.

3. A slide rule according to claim 1, comprising in addition a scale on each side of the slide representing values of E/( l from 10,000,000 to 40,000,000 for use when E is other than 30,000,000 and 0' is other than 0.3.

4. A slide rule according to claim 1, comprising in addition at least one scale on the thickness side of the slide representing values of OD/ID from 1.2 to 4.0 for use in determining thickness when OD/ID is other than 2. 

1. A slide rule for use in designing a Belleville spring, comprising two spaced elongated fixed members, a slide between said members and coextensive therewith, and a cursor mounted on the fixed members for movement along them, the parts of the rule having two sides on which there are the following scales: a. On one side of the rule scales for determining thickness in inches according to the formula: in which PF load at flat position E modulus of elasticity (30,000,000 for steel) sigma Poisson''s ratio q h/t h inside height, in inches t thickness OD outside diameter, in inches, in which formula steel values are substituted for material values and OD/ID 2 so that Alpha becomes a constant and sigma 0.3 E 3.0 X 107 M 0.685 Alpha 1.93 X 108 one of these scales representing values of PF from 1 to 10,000 extending from end to end of one of the fixed members, a second scale representing values of OD2 from 10 down to 0.1 on the slide adjacent and coterminous with the PF scale, a third scale representing values of t from 0.001 to 1.0 inch on the second fixed member and coterminous with PF and OD2 scales, a fourth scale on the slide representing values of q from 1.0 to 3 in the direction of 0.010 on the t scale and from 1.0 to 2 in the opposite direction and with the 1.0 value of q aligned with 0.0268 on the t scale if it is assumed that PF 100, OD 1 and q 1, and at least one scale on the slide representing values of OD/ID from 1.2 to 4, b. and on the other side of the rule scales for determining the stress at flat position according to the following formula: in which PF load at flat position OD outside diameter ScF compressive stress on convex side of the inner edge of flat, in p.s.i.
 2. A slide rule according to claim 1, comprising in addition at least one scale on the stress side of the slide representing values of OD/ID from 1.2 to 4, for use in determining stress OD/ID is other than
 2. 3. A slide rule according to claim 1, comprising in addition a scale on each side of the slide representing values of E/(1-sigma 2) from 10,000,000 to 40,000,000 for use when E is other than 30,000,000 and sigma is other than 0.3.
 4. A slide rule according to claim 1, comprising in addition at least one scale on the thickness side of the slide representing values of OD/ID from 1.2 to 4.0 for use in determining thickness when OD/ID is other than
 2. 